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February  2012, 32(2): 353-379. doi: 10.3934/dcds.2012.32.353

Lie's reduction method and differential Galois theory in the complex analytic context

David Blázquez-Sanz 1, and  Juan J. Morales-Ruiz 2,

1. 

Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia
2. 

Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain

Received  December 2010 Revised  April 2011 Published  September 2011

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.
Keywords: differential equations in the complex domain., Differential Galois theory.
Mathematics Subject Classification: Primary: 34M05; Secondary: 12H05, 14L99, 34A2.
Citation: David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353
References:
[1]

C. Athorne and T. Hartl, Solvable structures and hidden symmetries, J. Phys. A, 27 (1995), 3463-3474.  Google Scholar

[2]

C. Athorne, Symmetries of linear ordinary differential equations, J. Phys. A, 30 (1997), 4639-4649. doi: 10.1088/0305-4470/30/13/015.  Google Scholar

[3]

C. Athorne, On the Lie symmetry algebra of general ordinary differential equation, J. Phys. A, 31 (1998), 6605-6614. doi: 10.1088/0305-4470/31/31/008.  Google Scholar

[4]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58, Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010.  Google Scholar

[5]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem, J. Lie Theory, 20 (2010), 483-517. Google Scholar

[6]

R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'' Lectures at the R.G.I. in Park City, Utah, 1991. Google Scholar

[7]

A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'' Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986.  Google Scholar

[8]

J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'' Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000.  Google Scholar

[9]

J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258.  Google Scholar

[10]

J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math., 66 (2001), 67-87. doi: 10.1023/A:1010743114995.  Google Scholar

[11]

J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp.  Google Scholar

[12]

J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 43-69. doi: 10.1023/A:1013913930134.  Google Scholar

[13]

G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, (French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779.  Google Scholar

[14]

G. Casale, The Galois groupoid of Picard-Painlevé VI equation, "Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.  Google Scholar

[15]

G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité, Comment. Math. Helv., 83 (2008), 471-519. doi: 10.4171/CMH/133.  Google Scholar

[16]

G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'' (French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993.  Google Scholar

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W. Fulton and J. Harris, "Representation Theory. A First Course,'' Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.  Google Scholar

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A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965. Google Scholar

[19]

M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action, J. Math. Phys., 40 (1999), 3104-3122. doi: 10.1063/1.532749.  Google Scholar

[20]

J. E. Humphreys, "Linear Algebraic Groups,'' Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[21]

I. Kaplansky, "An Introduction to Differential Algebra,'' Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957.  Google Scholar

[22]

E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824. doi: 10.2307/2372550.  Google Scholar

[23]

E. R. Kolchin, "Differential Algebra and Algebraic Groups,'' Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973.  Google Scholar

[24]

A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures, J. Dynam. Control Systems, 1 (1995), 91-123. doi: 10.1007/BF02254657.  Google Scholar

[25]

S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten, Math. Ann. Bd., 25 (1885), 71-151. doi: 10.1007/BF01446421.  Google Scholar

[26]

S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'' Lepziger Berichte, 1893. Google Scholar

[27]

S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'' (German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971.  Google Scholar

[28]

S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235. Google Scholar

[29]

S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'' Reprinted by Chelsea books, N.Y., 1967. Google Scholar

[30]

J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités, J. Math. Pures Appl., 4 (1839), 423-456. Google Scholar

[31]

B. Malgrange, Le groupoïde de Galois d'un feuilletage, (French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.  Google Scholar

[32]

B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.  Google Scholar

[33]

B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'' (French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010. Google Scholar

[34]

P. Malliavin, "Géométrie Différentielle Intrinsèque,'' (French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972.  Google Scholar

[35]

M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases, Phys. Rev. A (3), 52 (1995), 936-940. doi: 10.1103/PhysRevA.52.936.  Google Scholar

[36]

J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., 8 (2001), 33-95, 97-111.  Google Scholar

[37]

J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

[38]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'' Progress in Mathematics, 179, Birkhäuser Verlag, Basel, 1999.  Google Scholar

[39]

J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884.  Google Scholar

[40]

K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173-179.  Google Scholar

[41]

K. Nishioka, General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1-6.  Google Scholar

[42]

K. Nishioka, Lie extensions, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85. doi: 10.3792/pjaa.73.82.  Google Scholar

[43]

K. Nomizu, "Lie Groups and Differential Geometry,'' The Mathematical Society of Japan, 1956.  Google Scholar

[44]

J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, (French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990.  Google Scholar

[45]

M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489.  Google Scholar

[46]

C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory], Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001.  Google Scholar

[47]

J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.   Google Scholar

[48]

J.-P. Serre, Espaces fibrés algebriques (French), Séminaire Claude Chevalley, 3 (1958), 1-37. Google Scholar

[49]

Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'' Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990.  Google Scholar

[50]

S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 25 (1984), 3155-3165. doi: 10.1063/1.526085.  Google Scholar

[51]

M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory, IEEE Trans. Automat. Control, 30 (1985), 266-272. doi: 10.1109/TAC.1985.1103934.  Google Scholar

[52]

H. Umemura, On the irreducibility of the first differential equation of Painlevé, in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988.  Google Scholar

[53]

H. Umemura, Galois theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1-58.  Google Scholar

[54]

H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59-135.  Google Scholar

[55]

H. Umemura, Sur l'équivalence des théories de Galois différentielles générales, (French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158.  Google Scholar

[56]

M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003.  Google Scholar

[57]

E. Vessiot, Sur l'intégration des equations différentielles linéaires (French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280.  Google Scholar

[58]

E. Vessiot, Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64.  Google Scholar

[59]

E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires, Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114. Google Scholar

[60]

E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33.  Google Scholar

[61]

E. Vessiot, Sur la théorie de Galois et ses diverses généralisations (French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85.  Google Scholar

[62]

E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif (French), Ann. École Norm. (3), 57 (1940), 1-60.  Google Scholar

show all references

References:
[1]

C. Athorne and T. Hartl, Solvable structures and hidden symmetries, J. Phys. A, 27 (1995), 3463-3474.  Google Scholar

[2]

C. Athorne, Symmetries of linear ordinary differential equations, J. Phys. A, 30 (1997), 4639-4649. doi: 10.1088/0305-4470/30/13/015.  Google Scholar

[3]

C. Athorne, On the Lie symmetry algebra of general ordinary differential equation, J. Phys. A, 31 (1998), 6605-6614. doi: 10.1088/0305-4470/31/31/008.  Google Scholar

[4]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58, Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010.  Google Scholar

[5]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem, J. Lie Theory, 20 (2010), 483-517. Google Scholar

[6]

R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'' Lectures at the R.G.I. in Park City, Utah, 1991. Google Scholar

[7]

A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'' Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986.  Google Scholar

[8]

J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'' Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000.  Google Scholar

[9]

J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258.  Google Scholar

[10]

J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math., 66 (2001), 67-87. doi: 10.1023/A:1010743114995.  Google Scholar

[11]

J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp.  Google Scholar

[12]

J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 43-69. doi: 10.1023/A:1013913930134.  Google Scholar

[13]

G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, (French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779.  Google Scholar

[14]

G. Casale, The Galois groupoid of Picard-Painlevé VI equation, "Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.  Google Scholar

[15]

G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité, Comment. Math. Helv., 83 (2008), 471-519. doi: 10.4171/CMH/133.  Google Scholar

[16]

G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'' (French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993.  Google Scholar

[17]

W. Fulton and J. Harris, "Representation Theory. A First Course,'' Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.  Google Scholar

[18]

A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965. Google Scholar

[19]

M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action, J. Math. Phys., 40 (1999), 3104-3122. doi: 10.1063/1.532749.  Google Scholar

[20]

J. E. Humphreys, "Linear Algebraic Groups,'' Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar

[21]

I. Kaplansky, "An Introduction to Differential Algebra,'' Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957.  Google Scholar

[22]

E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824. doi: 10.2307/2372550.  Google Scholar

[23]

E. R. Kolchin, "Differential Algebra and Algebraic Groups,'' Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973.  Google Scholar

[24]

A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures, J. Dynam. Control Systems, 1 (1995), 91-123. doi: 10.1007/BF02254657.  Google Scholar

[25]

S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten, Math. Ann. Bd., 25 (1885), 71-151. doi: 10.1007/BF01446421.  Google Scholar

[26]

S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'' Lepziger Berichte, 1893. Google Scholar

[27]

S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'' (German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971.  Google Scholar

[28]

S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235. Google Scholar

[29]

S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'' Reprinted by Chelsea books, N.Y., 1967. Google Scholar

[30]

J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités, J. Math. Pures Appl., 4 (1839), 423-456. Google Scholar

[31]

B. Malgrange, Le groupoïde de Galois d'un feuilletage, (French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.  Google Scholar

[32]

B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.  Google Scholar

[33]

B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'' (French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010. Google Scholar

[34]

P. Malliavin, "Géométrie Différentielle Intrinsèque,'' (French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972.  Google Scholar

[35]

M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases, Phys. Rev. A (3), 52 (1995), 936-940. doi: 10.1103/PhysRevA.52.936.  Google Scholar

[36]

J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., 8 (2001), 33-95, 97-111.  Google Scholar

[37]

J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120.  Google Scholar

[38]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'' Progress in Mathematics, 179, Birkhäuser Verlag, Basel, 1999.  Google Scholar

[39]

J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884.  Google Scholar

[40]

K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173-179.  Google Scholar

[41]

K. Nishioka, General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1-6.  Google Scholar

[42]

K. Nishioka, Lie extensions, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85. doi: 10.3792/pjaa.73.82.  Google Scholar

[43]

K. Nomizu, "Lie Groups and Differential Geometry,'' The Mathematical Society of Japan, 1956.  Google Scholar

[44]

J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, (French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990.  Google Scholar

[45]

M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489.  Google Scholar

[46]

C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory], Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001.  Google Scholar

[47]

J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.   Google Scholar

[48]

J.-P. Serre, Espaces fibrés algebriques (French), Séminaire Claude Chevalley, 3 (1958), 1-37. Google Scholar

[49]

Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'' Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990.  Google Scholar

[50]

S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 25 (1984), 3155-3165. doi: 10.1063/1.526085.  Google Scholar

[51]

M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory, IEEE Trans. Automat. Control, 30 (1985), 266-272. doi: 10.1109/TAC.1985.1103934.  Google Scholar

[52]

H. Umemura, On the irreducibility of the first differential equation of Painlevé, in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988.  Google Scholar

[53]

H. Umemura, Galois theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1-58.  Google Scholar

[54]

H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59-135.  Google Scholar

[55]

H. Umemura, Sur l'équivalence des théories de Galois différentielles générales, (French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158.  Google Scholar

[56]

M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003.  Google Scholar

[57]

E. Vessiot, Sur l'intégration des equations différentielles linéaires (French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280.  Google Scholar

[58]

E. Vessiot, Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64.  Google Scholar

[59]

E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires, Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114. Google Scholar

[60]

E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33.  Google Scholar

[61]

E. Vessiot, Sur la théorie de Galois et ses diverses généralisations (French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85.  Google Scholar

[62]

E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif (French), Ann. École Norm. (3), 57 (1940), 1-60.  Google Scholar

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